Opinion 40: Akalu Tefera: A Truly 3rd-Millennium Mathematics Ph.D.

By Doron Zeilberger

Written: March 10, 2000

Physical Science, at least for the last 200 years, had two kinds of research modes: theoretical and experimental. Now one can also add: computational. By contrast, mathematics, whether it was "pure" or "applied" only had the theoretical mode. No one got a Ph.D. for writing a computer program. Even a "conjectures-only" thesis was not acceptable. One had to prove theorems. If the theorems were too hard, one found easier versions, but proving theorems was the sine-qua-non for a math Ph.D.

Akalu Tefera, who expects to earn his Ph.D. this coming May, is a harbinger of a new kind of math thesis. It is not "experimental" in the semantic sense, since he did not compute zeros of the Riemann zeta function and study their distribution, or found new Mersenne primes. His results are rigorous theorems. But syntactically, it is analogous to an experimental thesis in physics or chemistry. There you spend the first two years building the apparatus, the next year or two in taking data, and then in the last year you analyze the data.

Akalu did a masterful job in implementing the Continuous mutli-WZ method, that was described in very broad outline by Herb Wilf and myself. After about two years, of successive improvements and enhancements, he "built the apparatus", his versatile packages Mint, and qMint (forthcomig).

After Akalu "built" (i.e. wrote) the "equipment" (i.e. software) he "took data" (i.e. used it to DISCOVER) the beautiful MULTI-VARIATE TEFERA INTEGRAL. This is a GENUINE theorem, even today, since, like the Selberg and Mehta integrals, it states something for an arbitrary dimension. Even Tefera's package, Mint, can't prove it in that generality. But for k=1,2,..., 6, Mint can find immediately, BEAUTIFUL WZ-Certificates. In the "analyzing the data" stage, Akalu detected a general pattern and proved, humanly, the general case. But without the equipment (the Maple package Mint), this would not have been possible. However, what's nice about Tefera's "equipment" is that it can be used on many other problems.

Two other 3rd-millennium Ph.D. thesis , that were completed ahead of schedule, way in the last millennium (c. 1998), are Frederic Chyzak's and Axel Riese's futuristic theses. I am sure that there are others, but we need many more, and in fifty years, this mode will be the norm.

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